A few days ago I asked this question on math.stackexchange:

Consider $(\tilde{M},g)$ a riemannian manifold and $M \subset \tilde{M}$ riemannian submanifold.

Is it true that if $M$ is a minimal submanifold of $\tilde{M}$ then for every $p \in M$ there exists a neighborhood $W$ of $p$ in $\tilde{M}$ such that $V=W\cap M$ has least area among every $\Omega \subset W$ with $\partial \Omega = \partial V$?

I've been thinking about it, I think it is true but I don't know how to prove.

If it's true, how should I go about proving it?

Link: math.stackexchange.com/questions/2365722

As asked there, we say a submanifold is minimal if the mean curvature vanishes identically, or equivalently, it's a critical point of the area functional.

Thanks in advance!